Source Mechanisms of Induced Earthquakes at The Geysers Geothermal Reservoir

LANE R. JOHNSON1

Abstract—At The Geysers geothermal reservoir in northern

California, evidence strongly suggests that activities associated

with production of electric power cause an increase in the number

of small earthquakes. First-degree dynamic moment tensors are

used to investigate the relationship between induced earthquakes

and injection of water into a well as part of a controlled experiment

in the northwest Geysers. The estimation of dynamic moment

tensors in the complex shallow crust at The Geysers is challenging,

so the method is described in detail with particular attention given

to the uncertainty in the results. For seismic events in the moment

magnitude range of 0.9–2.8, spectral moduli of dynamic moment

tensors are reliably recovered in the frequency range of 1–100 Hz,

but uncertainty in the associated spectral phases limits their use to a

few simple results. A number of different static moment tensors are

investigated, with the preferred one obtained from parameters of a

model fitted to the spectral modulus of the dynamic moment tensor.

Moment tensors estimated for a group of 20 earthquakes exhibit a

range of source mechanisms, with over half having significant

isotropic parts of either positive or negative sign. Corner frequen-

cies of the isotropic part of the moment tensor are about 40 %

larger than the average of the deviatoric moment tensor. Some

spatial patterns are present in source mechanisms, with earthquakes

closely related in space tending to have similar mechanisms, but at

the same time, some nearby earthquakes have very different

mechanisms. Tensional axes of displacement in the source regions

are primarily horizontal, while the pressure axes range from near

horizontal to vertical. Injection of water into the well in the center

of the study area clearly causes an increase in the number of

earthquakes per day, but an effect upon source mechanisms is not

evident.

Key words: Geysers geothermal reservoir,

induced earthquakes, source mechanism.

1. Introduction

The Geysers geothermal reservoir in northern

California is not only the world’s largest producer of

geothermal electricity but also the most thoroughly

studied example of induced seismicity related to

enhanced geothermal activity. Commercial produc-

tion of energy from reservoir steam began in the

1960’s, and since about 1970 the operators of the

reservoir have been injecting water back into the

reservoir. This injected water is steam condensate,

local rain, stream water, and more recently has

included treated wastewater from nearby communi-

ties. As is the case with most geothermal systems,

this injection of water is necessary in order to

maintain pressures and insure the long term economic

viability of the endeavor.

The Geysers geothermal reservoir is located in the

coast ranges of northern California and is contained

within the broad transcurrent boundary between the

North American and Pacific plates. There are several

active fault systems in the region (BUFE et al. 1981)

so seismic activity in the vicinity of the Geysers is

common (EBERHART-PHILLIPS and OPPENHEIMER 1984),

but within the geothermal reservoir seismicity does

not appear to be associated with any of these fault

systems. Rather, hypocenters of earthquakes within

the reservoir show a close association in both space in

time with either the production of steam or the

injection of water (BUFE et al. 1981; LUDWIN et al.

1982; EBERHART-PHILLIPS and OPPENHEIMER 1984;

OPPENHEIMER 1986; STARK 1992; ROMERO et al. 1994).

The general conclusion is that most of the seismicity

at The Geysers is induced by activities associated

with production of geothermal power (MAJER et al.

2007).

While there seems to be general agreement that

geothermal operations play a role in inducing seismic

activity at The Geysers, the physical mechanisms

involved in this process are still not understood.

Suggested mechanisms include volumetric contrac-

tion of the reservoir caused by the withdrawal of fluid

1 Earth Sciences Division, Lawrence Berkeley National

Laboratory, Berkeley, CA 94720, USA. E-mail:

LRJohnson@lbl.gov

Pure Appl. Geophys. 171 (2014), 1641–1668

� 2014 The Author(s)

This article is published with open access at Springerlink.com

DOI 10.1007/s00024-014-0795-x Pure and Applied Geophysics

(MAJER and MCEVILLY 1979) or thermal cooling

(DENLINGER et al. 1981; STARK 2003), conversion of

aseismic creep to stick-slip failure (ALLIS 1982), and

the increase in the coefficient of friction caused by

precipitation of silica on fracture surfaces (ALLIS

1982). A more complete understanding of the cause

of induced earthquakes at The Geysers could help

guide the economic development of the reservoir and

help alleviate concerns in surrounding communities.

An objective of the present study is to take

another look at the cause of induced seismicity at The

Geysers through a careful analysis of seismograms

recorded by a local network. Source mechanisms of a

group of earthquakes that occurred during a con-

trolled experiment are characterized by estimating

their dynamic force moment tensors. The interpreta-

tion of these moment tensors can help constrain the

physical processes taking place in the source regions

of the earthquakes.

2. Previous Studies

Before describing the methods and results of the

present study, it is useful to briefly review some

previous studies related to source processes of

earthquakes and material properties of the shallow

crust at The Geysers.

A number of geodetic studies have attempted to

place constraints on deformation that is taking place

within the shallow crust at The Geysers. The reasons

for this deformation in a vapor dominated reservoir

such as The Geysers are complicated and involve an

interplay between several processes that accompany

two-phase fluid flow in a fractured medium with both

pressure and temperature variations (PRUESS and

NARASIMHAN 1982). DENLINGER et al. (1981) analyzed

geodetic measurements and found that the geother-

mal reservoir is subsiding. LOFGREN (1981) and

DENLINGER and BUFE (1982) analyzed local geodetic

data at The Geysers and found a negative strain in

horizontal directions and subsidence in the vertical

direction. MOSSOP and SEGALL (1997, 1999) and

SEGALL and FITZGERALD (1998) analyzed GPS and

leveling data that confirmed earlier estimates of

subsidence within the reservoir, with rates as high as

0.047 ± 0.002 m/year during the period 1977–1996.

They also estimated volume strain, temperature

change, pressure change, and bulk modulus in the

producing part of the reservoir. PRESCOTT and YU

(1986) used regional geodetic data to show that

regional strain in The Geysers was primarily a shear

with extension oriented N79�W and contraction ori-

ented N11�E. In summary, deformation at The

Geysers describes a geothermal reservoir that is

undergoing volumetric contraction while contained

within a regional shear field.

Seismological studies at The Geysers have used a

number of different approaches to infer forces and

processes acting in the source regions of earthquakes.

OPPENHEIMER (1986) performed fault plane studies

using polarity of p waves for 210 seismic events at

The Geysers and found below a depth of 1 km pri-

marily horizontal extension in a WNW direction and

compression ranging from horizontal to vertical along

a NNE azimuth. The general conclusion was that

source mechanisms were consistent with the regional

strain field.

O’CONNELL and JOHNSON (1988) calculated

dynamic moment tensors of three events at The

Geysers and obtained results that were more robust

than those based on first motion polarities. One of the

events appeared to have a significant isotropic part,

but the possibility that this was an artifact caused by

errors in the source location or velocity model used in

the inversion could not be ruled out.

KIRKPATRICK et al. (1996) obtained static moment

tensors for 2,552 events at The Geysers, with 1,697 of

the events in the northwest Geysers, the location of

the present study. Amplitudes of direct p and s phases

were measured and Green functions calculated with

ray theory were used to solve for static moment

tensors. About half of the events were predominantly

double couple, but the rest contained significant iso-

tropic and compensated linear vector dipole fractions.

The isotropic parts were both positive and negative in

sign with the fraction of the total moment ranging up

to 0.3.

ROSS et al. (1996, 1999) measured polarities of p

and s waves together with p/s amplitude ratios and

used linear programming methods (JULIAN 1986;

JULIAN and FOULGER 1995) to obtain relative static

moment tensors for 30 events at The Geysers. About

70 % of the events had significant isotropic parts,

1642 L. R. Johnson Pure Appl. Geophys.

both positive and negative in sign, with the isotropic

fraction of the total moment being[0.2 for 50 % and

[0.3 for 17 %. Orientations of the T and P axes

showed considerable variation.

GUILHEM et al. (2014) compared static moment

tensor solutions obtained with a short-period local

network at The Geysers with those obtained with a

long-period regional network and found generally

consistent results for 14 events. A time-domain

method (DREGER 2003) was used to obtain static

moment tensors for two different frequency bands

and these were compared with first-motion solutions.

Decompositions of the moment tensors varied

somewhat between the different solutions, but on the

average the events were about half double couple, the

isotropic fraction ranging between 0 and 0.3, and the

remaining fraction was compensated linear vector

dipole.

Extracting source information from seismic

waveform data requires the calculation of elastic

wave propagation effects and, this in turn, requires

knowledge about the velocity structure. Considerable

information of this type is available, and all of it

indicates that the upper crust at The Geysers is highly

heterogeneous. This begins with the topography, as

the terrain is rugged and mountainous with elevations

ranging from\500 m to over 1,400 m. Because of the

steep terrain, there are numerous landslides. The

geology is equally complex, with the Franciscan

subduction complex broken into a series of fault-

bounded slabs that are intruded by a silicic batholith

(THOMPSON 1992). The geothermal reservoir is highly

fractured with the orientation of the fractures gener-

ally random (THOMPSON and GUNDERSON 1992; BEALL

and BOX 1992). The general reservoir model has

high-angle fractures of varying strike separating the

sedimentary section into blocks that contain low-

angle fractures of limited extent, while the intrusive

section is dominated by high-angle fractures.

O’CONNELL and JOHNSON (1991) developed a one-

dimensional velocity model for The Geysers through

a process of progressive inversion for p velocity, s

velocity, and hypocenters. Particular attention was

given to the compressional velocity to shear velocity

ratio vp/vs that shows considerable variation with

depth and a minimum that correlates with the zone of

maximum steam production. BOITNOTT (1995) used

laboratory measurements on rocks from The Geysers

and the Biot-Gassmann poroelastic theory to show

that the variations in vp/vs are consistent with changes

in the degree of saturation. ROMERO et al. (1994) also

used a progressive inversion procedure to solve the

three-dimensional problem for hypocenters, p veloc-

ity, vp/vs velocity ratio, and a differential quality

factor for p waves Qp. The three-dimensional varia-

tions in velocity and attenuation appeared to correlate

with the geology.

ZUCCA et al. (1994) used measurements of arrival

times and pulse widths to calculate a three-dimen-

sional model for compressional velocity vp and a one-

dimensional model for compressional attenuation Qp

down to a depth of 1.5 km below sea level. The

shallow part of the vp model correlated with surface

geology, and, at depth, low values of vp correlated

with parts of the reservoir where there had been

production of steam and reduction in pressure. The

Qp values are low near the surface, increase with

depth, and then decrease within the zone of steam

production.

JULIAN et al. (1996) used arrival times of direct

compressional and shear waves to calculate three-

dimensional models of vp and the ratio vp/vs from the

surface down to a depth of 3 km below sea level. The

lateral variation in vp was ±12 % of mean values and

the pattern was in general agreement with that of

ZUCCA et al. (1994). The most notable feature of the

vp/vs results was a strong -9 % anomaly that corre-

lated with the most intensively exploited part of the

geothermal reservoir.

ROMERO et al. (1997) used a spectral ratio method

to estimate three-dimensional variations in differen-

tial attenuation for both p and s waves in the

northwest Geysers. The differential attenuation

showed significant lateral variations, particularly near

the surface where it was correlated with mapped

geologic units. A zone of high p wave attenuation and

low vp/vs appeared to delineate the geothermal

reservoir.

MAJER et al. (1988) used vertical seismic profiling

to study anisotropy in the upper 1.5 km of The

Geysers reservoir. They found no evidence for

anisotropy in p waves, but sv waves had about 11 %

higher velocities than sh waves. They concluded that

this was consistent with effects of known fracture

Vol. 171, (2014) Source Mechanisms of Induced Earthquakes 1643

geometry. EVANS et al. (1995) used shear-wave

splitting to study seismic anisotropy at The Geysers

and found a complex pattern. If the anisotropy is

confined to the upper 1.5 km of the crust, it averages

4 %. ELKIBBI et al. (2005) and ELKIBBI and RIAL

(2005) used shear-wave splitting to study anisotropy

in both the northwest and southeast Geysers and also

found the average anisotropy in shear wave velocities

to be about 4 %. They concluded that the results in

the northwest Geysers could be explained by systems

of vertical extensional cracks aligned parallel to the

direction of principal horizontal compressive stress

(N to NE). In the southeast Geysers the fast shear

wave polarizations showed more variation and sug-

gested that more than one system of fractures might

be present.

The possibility that material properties at The

Geysers may be changing with time because of pro-

duction activities must also be considered. FOULGER

et al. (1997) found that between 1991 and 1994 the

vp/vs ratio in the center of the reservoir decreased by 4

% and attributed this to a change in vp caused by a

conversion of pore water to steam.

The picture that emerges from this consideration of

previous studies at The Geysers is one of a fairly com-

plicated shallow crust. There are strong lateral

variations in topography and geology at the surface that

cause similar variations in elastic velocities. These lat-

eral variations in elastic velocities extend to a depth

where they correlate with lithology as well as steam

production activities. There is also evidence for signif-

icant vertical and lateral variations in the attenuation and

anisotropy of elastic waves. Superimposed on this het-

erogenous crustal structure is a deformation field that

involves volumetric contraction within a regional shear

field. Earthquakes at The Geysers are more complicated

than most tectonic earthquakes, as they contain signifi-

cant isotropic parts and are related to steam production

activities. The situation is further complicated by the

likelihood that both material properties and earthquake

mechanisms are functions of time.

3. Data

The seismic data used in this study are selected to

take advantage of a recent controlled experiment that

is investigating the relationship between seismicity

and the injection of water. The Northwest Geysers

Enhanced Geothermal System Demonstration Project

began in 2010 as a test of the feasibility of stimu-

lating a deep high-temperature reservoir (RUTQVIST

et al. 2010; GARCIA et al. 2012). The project is

located in an undeveloped part of the reservoir where

two previously abandoned wells, Prati 32 (P32) and

Prati State 31 (PS31), were prepared for use as an

injection and production well pair. P32 was com-

pleted to a depth of 2,672 m below sea level (bsl).

The injection interval in P32 is between 1,653 and

2,672 m bsl. A square study area centered on P32

with dimensions of 1.5 km is the site selected for this

study. Only earthquakes with epicenters within this

study are used. Figure 1 is a map showing the loca-

tion of this study area within The Geysers geothermal

reservoir along with the locations of the seismo-

graphic stations that are used.

The observational data for this study are the

seismograms produced by the LBNL seismic moni-

toring network at The Geysers. This network consists

of three-component digital stations that are mostly

within the production boundaries at the Geysers

-122.95 -122.90 -122.85 -122.80 -122.75 -122.70 -122.65 38.70

38.75

38.80

38.85

38.90

Longitude, deg

Latit

ude,

deg

ACRAL1

AL2

AL3

AL4AL5

AL6

BRP

BUC

CLV

HBW

HER

JKR

LCK

MCL

RGP

SB4STY

SQK

FUMDRK

DXR

Figure 1Map of The Geysers geothermal reservoir with the dashed line

showing the approximate boundary of the steam production zone.

The small circles are locations of stations of the LBNL seismic

monitoring network that are used in this study. The rectangle in the

northwest part of the reservoir is the study area that contains the

epicenters of all events used in this study

1644 L. R. Johnson Pure Appl. Geophys.

(MAJOR and PETERSON 2007). The seismometers have

a response that is flat to ground velocities at fre-

quencies above 4.5 Hz. The seismometer outputs are

continuously digitized with a resolution of 24 bits and

a sample rate of 500 samples per second. The data are

routinely processed to identify seismic events along

with their hypocenters and magnitudes with a lower

threshold for complete coverage at the Geysers of

about magnitude one. Locations of the LBNL seis-

mographic stations used in this study are shown in

Fig. 1. All of the stations used in this study are

located at the free surface.

As part of the EGS Demonstration Project,

injection of water into P32 began on October 6, 2011

(GARCIA 2012). An initial rate of 1,100–1,200 gallons

per minute (gpm) was used to collapse the steam

bubble in the well bore and continued for 12 h. The

rate was then reduced to 400 gpm and maintained for

54 days. On 29 November, 2011, the injection rate

was increased to 1,000 gpm, on 7 March, 2012, it was

reduced to 700 gpm, and on 11 June 2012, it was

reduced to 400 gpm. This injection rate is plotted in

Fig. 2. Note that water is simply poured into the well

with no pressure applied other than the force of

gravity.

The frequency of earthquakes that have epicenters

in the study area around P32 for a time interval of 600

days is shown in Fig. 2. In the 300 days before

injection began, there are 241 events in the magnitude

range 0.4 B M B 1.9, while in the 300 days after

injection began there are 2,107 events in the magni-

tude range 0.0 B M B 2.8. Figure 2 shows that there

is a clear correlation between the rate of seismicity

near P32 and the rate of water injection into that well.

Prior to the beginning of injection, the rate of seis-

micity has a low background value of about 1–2

events per day, but this jumps to a much higher value

when injection begins. Abrupt increases in injection

rate cause abrupt increases in seismicity.

From the earthquakes shown in Fig. 2, a group of

20 are selected for moment tensor inversion, with the

times of these events marked at the top of the figure.

The selection criteria are designed to provide a broad

range in event times, depths, and magnitudes. The

group includes five events spread out over the time

interval before injection in P32 began and 15 events

that sampled all of the time intervals of different

injection rate shown in Fig. 2. Depths of the selected

events ranged between about 2.0 and 3.0 km below

sea level. Magnitudes ranged between 0.9 and 2.8,

with the lower limit set by the requirement of a good

signal to noise ratio on the seismograms and the

upper limit equal to the largest event in the time

interval considered. The group of selected events is

listed in Table 1.

Hypocenters of the 20 events selected for moment

tensor inversion are plotted in the map of Fig. 3 and

cross sections in Figs. 4 and 5. These figures show

the proximity of the events to the P32 well. All events

are within 720 m horizontal distance of P32 or the

downward projection of its bottom. Events with

numbers 1–5 occurred before injection into P32

began, and they are at approximately the same depth

with their epicenters roughly aligned in a NNE

direction. Events with numbers 6–20 occurred while

injection into P32 was taking place and their hypo-

centers show significant clustering about the well for

many but not all of the events.

The events listed in Table 1 are analyzed with

data recorded at the stations shown in Fig. 1. More

stations than those shown in Fig. 1 are available, but

stations at epicentral distances greater than about two

-300 -200 -100 0 100 200 300 0

5

10

15

20

25

30

35

40

Time, days

Eve

nts/

day

0

200

400

600

800

1000

1200

Inje

ctio

n ra

te, g

al/m

in

Figure 2Number of earthquakes per day with Mw C 0 and epicenters in the

study area shown in Fig. 1. Time is measured in days from the

beginning of injection on 6 October 2011. The dashed line is the

rate of injection of water into P32. The symbols at the top denote

origin times of events in Table 1

Vol. 171, (2014) Source Mechanisms of Induced Earthquakes 1645

source depths are not used because the seismograms

at these distances are dominated by surface waves

that are difficult to model with Green functions. The

strong three-dimensional heterogeneity in topography

and material properties near the surface at The Gey-

sers is likely to cause strong scattering and multi-

pathing of surface waves. Although data from all of

the stations shown in Fig. 1 are not available for all of

Table 1

Origin times, hypocenters, and magnitudes of events selected for moment tensor inversion

Event no. Years Months Days Hours Minutes Seconds Latitude �N Longitude �E Depth (m) Mw

1 2011 3 8 21 22 23.28 38.837970 -122.837750 2,194 1.78

2 2011 4 25 8 31 2.08 38.842730 -122.832750 2,276 1.53

3 2011 5 3 14 2 31.84 38.837880 -122.835340 2,399 1.73

4 2011 7 12 22 8 32.20 38.834890 -122.838560 2,551 1.04

5 2011 8 3 23 45 46.66 38.836550 -122.838220 2,601 1.56

6 2011 10 7 2 5 25.92 38.838370 -122.838520 2,728 0.93

7 2011 10 7 11 48 51.32 38.837980 -122.839990 1,999 1.82

8 2011 10 10 7 9 3.18 38.838600 -122.839870 2,745 1.45

9 2011 10 26 13 55 34.03 38.843080 -122.841790 2,120 2.30

10 2011 11 14 6 30 51.17 38.840660 -122.840990 2,850 1.74

11 2011 11 30 3 34 48.11 38.839690 -122.839340 2,560 1.66

12 2011 12 9 13 41 48.06 38.835170 -122.841280 2,417 2.45

13 2011 12 28 6 25 38.90 38.838290 -122.839870 2,909 2.40

14 2012 1 6 20 55 39.88 38.839580 -122.837520 2,700 2.75

15 2012 1 29 15 11 49.28 38.842200 -122.837460 2,020 1.42

16 2012 2 29 17 30 54.31 38.842780 -122.840770 2,831 2.12

17 2012 3 9 19 29 8.59 38.841460 -122.837850 2,574 2.17

18 2012 4 21 9 38 16.26 38.838810 -122.840060 2,821 1.50

19 2012 5 31 5 31 23.90 38.840180 -122.838060 2,940 2.82

20 2012 6 28 12 49 5.56 38.840180 -122.837300 2,535 2.14

Depths of hypocenters are measured from sea level. The magnitude Mw is obtained from the Euclidean norm of the scalar moment tensor

-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Y (

km)

X (km)

1

2

3

4

5

6 7

8

9

10

11

12

13

14

15

16

17

18

19 20

P32

N

SW E

Figure 3Map of epicenters of events selected for moment tensor inversion.

Numbers correspond to the event numbers in Table 1. The line is

the surface projection of the P32 well with the label at the well

head

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

X (km)

Z (

km)

1 2

3 4

5 6

7

8

9

10

11 12

13

14

15

16

17

18 19

20

P32W E

-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Figure 4Cross section from west to east through the center of the map in

Fig. 3. Z is depth below sea level

1646 L. R. Johnson Pure Appl. Geophys.

the events listed in Table 1, data from a minimum of

ten stations are available for each event. Only stations

with three operating components are used, so at least

30 components of data are used in each moment

tensor inversion.

4. Moment Tensor Method

The basic procedure followed in this study is to

characterize an earthquake by its first-degree

dynamic moment tensor, which is a general and fairly

complete mathematical representation of an earth-

quake source. The general idea is that effects of wave

propagation through the earth that are present in the

observational data will be removed in the process of

estimating the moment tensor. While this ideal can

not be completely realized in a complicated region

such as The Geysers, properties of the source process

are much more clearly displayed in the moment

tensor than in the original seismograms.

Consider the situation where K different seismo-

gram components are available and let the

k component that is recorded at location x and time

t be denoted by vk(x, t). This component is sampled

in time with a sampling interval Dt and a data win-

dow is selected beginning before the arrival of the

first p wave, continuing to a couple of seconds after

the arrival of the direct s wave, and having a total of

N samples. A discrete Fourier transform is performed

on this data window to obtain Vk(x, f), where upper

case letters are used to denote functions in the fre-

quency domain. The presence of random noise in the

data is taken into account by calculating the variance

of each component var[vk(x, t)] from an interval of

time immediately before the arrival of the direct p

wave. It is assumed that the noise is stationary,

uncorrelated between components, and independent

of frequency. Then, using Parseval’s formula, the

variance of the data in the frequency domain is

var½Vkðx; f Þ� ¼ NðDtÞ2var½vkðx; tÞ�: ð1Þ

The method of estimating the moment tensor in

this study follows that of STUMP and JOHNSON (1977).

It is assumed that the earthquake source process can

be adequately represented by the symmetric first-

degree moment tensor mij(t), which is equivalent to

assuming that the wavelengths of interest are large

compared to the dimensions of the source volume.

This assumption is reasonable at frequencies up to

about the corner frequency of the source spectrum

(STUMP and JOHNSON 1982), but caution must be

exercised in the interpretation of frequencies beyond

this point. Given this assumption, the k seismogram

component can be represented

vkðx; tÞ ¼ rðtÞ�gki;jðx; t; 0; 0Þ�mijðtÞ; ð2Þ

where r(t) is the instrument response function of the

seismograph, gki,j is the j spatial derivative of the

Green function, and mij(t) is the dynamic force

moment tensor. Here the symbol � represents con-

volution in time, and it is assumed that the source

process is centered at the origin of the coordinate

system and starts at time t = 0. In this and later

equations a repeated index in a product of terms

implies summation from 1 to 3. Because mij(t) is a

symmetric second-rank tensor, it has six independent

values at each instant of time.

The seismogram in the frequency domain is

Vkðx; f Þ ¼ Rðf ÞGki;jðx; f ; 0; 0ÞMijðf Þ: ð3aÞ

Such an equation can be written for every seismo-

gram component that is observed, with only the

Green function changing as the component and

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Y (km)Z

(km

)

1 2

3 4 5 6

7

8

9

10

11 12

13

14

15

16

17

18 19

20

P32S N

-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Figure 5Cross section from south to north through the center of the map in

Fig. 3. Z is depth below sea level

Vol. 171, (2014) Source Mechanisms of Induced Earthquakes 1647

location of the seismogram changes. Thus, if K dif-

ferent seismogram components are available, at every

frequency there is a system of K linear equations that

can be solved for the six independent unknown values

of Mij(f). The dynamic moment tensor mij(t) will

generally contain a static offset so its Fourier trans-

form Mij(f) will diverge as f-1 at low frequencies.

Thus, the numerical stability of the solution is

improved by considering the alternative system

Vkðx; f Þ ¼ Rðf Þi2pf

Gki;jðx; f ; 0; 0Þ _Mijðf Þ; ð3bÞ

where _Mijðf Þ and its inverse Fourier transform _mijðtÞ is

the dynamic moment rate tensor. The statistical reli-

ability of the problem can be improved by weighting

the data with the inverse standard deviation of the

noise, and, in that case, the system becomes

Vkðx; f Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

var½Vkðx; f Þp

�¼ Rðf Þ

i2pf

Gki;jðx; f ; 0; 0Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

var½Vkðx; f Þ�p

_Mijðf Þ: ð3cÞ

With K components of data available for analysis,

at any given frequency value f the Fourier transform

values (with possible weighting) can be collected to

form a column matrix Vðf Þ with dimensions K by 1.

For each component, there are six Green functions

corresponding to the six independent terms of the

moment tensor and these are treated similarly to the

data to form a matrix Gðf Þ that has dimensions K by 6,

where the instrument response and possible weighting

are included with the Green function. The unknown

six independent terms of the moment rate tensor form

the column matrix _Mðf Þ with dimensions 6 by 1.

Thus, for each frequency any of Eqs. 3a and b, or c can

be written as the linear matrix equation

Vðf Þ ¼ Gðf Þ _Mðf Þ: ð4Þ

This is a linear inverse problem, and the solution can

be written formally as

_Mðf Þ ¼ G�1ðf Þ Vðf Þ: ð5Þ

The variance of the solution is

var½ _Mðf Þ� ¼ ðG�1ðf ÞÞ2 var½Vðf Þ�: ð6Þ

Following STUMP and JOHNSON (1977), it is convenient

to obtain the solution to this inverse problem using

singular value decomposition (LANCZOS 1961).

The approach outlined in the preceding para-

graphs produces estimates of _Mijðf Þ and var½ _Mijðf Þ�:An inverse Fourier transform leads to _mijðtÞ and a

numerical integration then leads to the dynamic

moment tensor mij(t). As the first step in the inter-

pretation of the moment tensor, it is convenient to

decompose it into its eigenvalues and eigenvectors.

Because the moment tensor is symmetric, the eigen-

values are real and can be denoted

KTðtÞ�KIðtÞ�KPðtÞ: ð7Þ

The corresponding unit eigenvectors are

eTðtÞ; eIðtÞ; ePðtÞ: ð8Þ

The directions of the eigenvectors are referred to as

the T, I, and P axes and interpreted as principal

directions of displacement in the source region.

The trace of the moment tensor is an invariant that

is related to the isotropic part of the moment tensor

by

misoðtÞ ¼1

3

h

KTðtÞ þ KIðtÞ þ KPðtÞi

¼ 1

3

h

m11ðtÞ þ m22ðtÞ þ m33ðtÞi

ð9Þ

and identified with volume change in the source

process. The deviatoric moment tensor is defined as

m0ijðtÞ ¼ mijðtÞ � misoðtÞdij: ð10Þ

The static moment tensor is defined as

~mij ¼ mijðt!1Þ ¼ _Mijðf ! 0Þ ð11Þ

and is equivalent to a dynamic moment tensor with

all elements having the same time dependence of a

unit step function. In practice, the static moment

tensor is often estimated at a time other than infinity

or a frequency other than zero, so a variety of dif-

ferent results are possible for the same event. It is also

possible to assume that all elements of the moment

tensor have the same functional time dependence that

is different from a step function and include the

estimation of this function in the process of estimat-

ing the static moment tensor. This static moment

tensor can also be decomposed into eigenvalues and

eigenvectors that are no longer a function of time or

frequency.

1648 L. R. Johnson Pure Appl. Geophys.

The scalar moment mo is defined as the norm of

the static moment tensor, which is here taken to be

the Euclidean norm and equal to the maximum of the

absolute values of the eigenvalues (FRANKLIN 1968).

mo ¼ maxn

j ~KT j; j ~KI j; j ~KPjo

: ð12Þ

BOWERS and HUDSON (1999) discuss alternative defi-

nitions of the scalar moment. The moment magnitude

is given by

Mw ¼2

3log10ðmoÞ; ð13Þ

where mo is given in units of GNm. All moments in

this study are given in units of GNm.

The velocity model used in calculating Green

functions plays a critical role in the estimation of the

moment tensor. The complicated shallow crust at The

Geysers presents a problem in this respect because of

the difficulty of calculating accurate and complete

Green functions in the frequency range 0–250 Hz for

a model that is heterogeneous in three dimensions.

The choice made in the present study is to use a one-

dimensional model that allows straightforward Green

function calculations and then make adjustments to

approximate some of the three-dimensional effects.

The chosen base model is a layered model that was

developed by O’CONNELL and JOHNSON (1991) for The

Geysers region, and it is shown in Fig. 6.

The model shown in Fig. 6 is a base model that is

modified to make it more appropriate for the individ-

ual seismic stations. This model has two layers above

sea level, with the upper layer having a thickness of

200 m and the second layer having a nominal thick-

ness of 1,000 m, so the surface elevation has a nominal

value of 1,200 m. There is considerable topography at

The Geysers, with the elevation of the seismic stations

shown in Fig. 1 ranging between 327 and 1,112 m

above sea level. To accommodate this range in ele-

vation, the model is adjusted to have the correct

elevation at each station by adjusting the thickness of

the second layer. The compressional and shear veloc-

ity of the second layer are also adjusted at each station

so as to minimize travel time residuals that result when

origin times and hypocenters of the seismic events are

determined. Thus, a different model is used for each

seismic station with the differences between stations

confined to the two layers above sea level. Velocities

above sea level range between 1,800 and 3,300 m/s for

p waves and between 1,400 and 2,060 m/s for s waves.

These near surface adjustments achieve the same

effect as commonly used station correction terms, but

have the advantage of correctly including elevation

and slowness effects.

Attenuation is included in the model by assuming

quality factors of Qp = Qs = 100 above sea level and

Qp = Qs = 200 below sea level. These values are

chosen to be somewhat larger than the one-dimen-

sional estimates of ZUCCA et al. (1994) and ROMERO

et al. (1997) in order to avoid a problem caused by

the lateral variations in attenuation at The Geysers.

As pointed out by ROMERO et al. (1997), there are

significant differences in the frequency content of the

same seismic phase observed at different stations.

Thus, if Green functions are calculated with average

attenuation values, the energy from seismic stations

with enriched frequency content will contaminate the

moment tensor estimates at high frequencies. Part of

this problem may relate to the possibility that much

of the observed attenuation may be caused by scat-

tering, while the Green functions are calculated

assuming it is caused by intrinsic attenuation.

The Green functions are calculated for a one-

dimensional layered model using the invariant

-2000 -1000 0 1000 2000 3000 4000 5000 6000 0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

6000

Depth, m

Vel

ocity

, Den

sity

Figure 6Compressional velocity (solid line) and shear velocity (short dash

line), both in units of m/s, and density (long dash line) in units of

kg/m3

Vol. 171, (2014) Source Mechanisms of Induced Earthquakes 1649

embedding method of KENNETT (1974) and KENNETT

and KERRY (1979). The dispersion associated with

attenuation is included with the constant Q formula-

tion of KJARTASSON (1979).

5. Dynamic Moment Tensor Results

In this study the data sample interval is Dt ¼ 2

m s and the data window has N = 2,048 sample

points for a total duration of 4.1 s, with the first p

wave arriving at least 0.5 s after the beginning of the

window. The Nyquist frequency is fN = 250 Hz. A

time window of 0.4 s duration before the arrival of

the p wave is used to estimate the noise level

var[vk(x, t)]. A cosine taper is applied to the second

half of the time window to bring the amplitude

smoothly to zero at the end.

The events shown in Fig. 2 are located with a

three-dimensional model using arrival times of p

waves and s waves (Ernie Majer, personal com-

munication, 2012). However, the model used in the

moment tensor inversion process is one-dimen-

sional. Thus, new origin times and hypocenter

locations are determined for each event in Table 1

using the model of Fig. 6 and employing travel

times of both p and s waves. Differences between

the original and relocated hypocenters are small,

generally \100 m, but this step helps improve

compatibility between the observational data and

the Green functions. The standard errors of the

travel time residuals are always \0.02 s and the

estimated uncertainty in relocated hypocenters is

about 50 m in the horizontal direction and 100 m

in the vertical direction.

The basic procedure used in estimating dynamic

moment tensors along with some of the problems

that are encountered in the process are best illus-

trated by showing some of the intermediate results

for one of the events. Figure 7 shows observational

data at one station for event 12 of Table 1. Also

shown for comparison are data calculated with a

static moment tensor from Table 2 that is combined

with the Green function and instrument response. It

is assumed in this case that all elements of the

moment tensor have the same time function, which

in this case is a simple pulse with a corner frequency

of 10 Hz and high-frequency asymptote power of 3.

The observed data have portions of time with sig-

nificantly larger amplitudes than the calculated data,

such as the time before the arrival of the s wave on

the T component and the time beginning about 0.5 s

after the s wave on all three components. This lack

of agreement raises a critical question that impacts

several aspects of this study. Namely, are these un-

modeled parts of the observed data caused by the

time functions of the dynamic moment tensor or are

they artifacts due to inaccuracies in the Green

functions?

As described in the previous section, the dynamic

moment tensor is obtained as the solution to a linear

inverse problem that produces the Fourier transform

of the moment rate function _Mðf Þ (Eq. 5). The

spectral modulus of that solution is displayed for the

six independent elements of the moment rate function

in Fig. 8. Also shown in Fig. 8 is the uncertainty in

the modulus of the moment rate tensor from Eq. 6.

Over most of the frequency range, the spectral

modulus is much larger than its uncertainty and, thus,

well determined, but this not true at both the lowest

0 1 2 3 4

Time (sec)

R 0.63E+05

0.25E+05

T 0.56E+05

0.45E+05

Z 0.47E+05

0.21E+05

Figure 7Observed seismograms in the radial (R), transverse (T), and vertical

(Z) direction at station AL1 at an epicentral distance of 3.67 km

and azimuth of 275� from event 12. The 2nd, 4th, and 6th traces are

calculated seismograms that result when a static moment tensor is

combined with the Green function, source pulse, and instrument

response. The numbers on the right are maximum amplitudes in

digital counts. The zero time is 0.67 s after the origin time of the

event. The dashed vertical lines show the arrival times of direct p

and s waves calculated with ray theory

1650 L. R. Johnson Pure Appl. Geophys.

and highest frequency ranges. At frequencies above

100 Hz, the spectral modulus is basically noise. This

is caused by the fact that the source time functions of

the moment tensor elements all have corner fre-

quencies in the range of 10–20 Hz and at higher

frequencies the amplitude falls off rapidly until the

noise level is reached at about 100 Hz. This aspect of

the solution is not much of a problem because, if

desired, the frequencies above 100 Hz could be

removed by filtering without affecting any of the

lower frequencies.

The fact that the spectral modulus of the moment

rate tensor elements begins to approach its uncertainty

at frequencies \1 Hz is a more serious problem. The

most likely cause of this problem is the instrument

response that has a corner frequency at 4.5 Hz and a

fall-off power of -3 below that so the lowest fre-

quencies are weakly recorded in the data. Also note that

uncertainties shown in Fig. 8 are calculated assuming

the noise spectrum is independent of frequency, so if

this is not true and the noise is enriched in low fre-

quencies, then the situation is even worse than shown in

Fig. 8. Because of this problem at low frequencies,

features of the dynamic moment tensors at frequencies

\1 Hz are regarded as having considerable uncertainty

and used only when necessary. Their use can not be

completely avoided because moment rate tensors in the

time domain must be integrated to obtain moment

tensors and low frequencies play an essential role in

this integration process.

In analyzing spectral moduli of the moment rate

tensors such as those shown in Fig. 8, it is useful to

parameterize the primary features of the spectrum. A

model (IHAKA 1985) is fitted to the spectral modulus

of each moment rate tensor element that has the form

j _Mijðf Þj ¼~mij

1þ ffc

� �shþ n: ð14Þ

This model is fitted to the spectral modulus in the

frequency interval between 1 and 100 Hz, with lower

and higher frequencies not included because of the

noise problems discussed earlier. The short dash lines

in Fig. 8 are this model fitted to the spectral moduli

and plotted over the entire frequency range.

In this spectral model the low frequency level of

the spectral modulus is interpreted as an estimate of

the static moment tensor ~mij (Eq. 11). The parameter

fc is an estimate of the corner frequency of the source

time function with sh an estimate of the power of the

high frequency asymptote. The parameter n is an

estimate of the average random noise level in the

spectral domain and generally has a small value that

has not proved to be useful. The estimate of the static

moment tensor ~mij is essentially a measure of the

level of the spectral modulus of the moment rate

tensor in the frequency interval between 1 Hz and the

corner frequency fc that is typically in the range

10–20 Hz. Accordingly, an estimate of the uncer-

tainty of this static moment tensor element is the root-

mean-square value of the difference between the

observed and fitted spectral estimates in this same

frequency interval. Here, and throughout this paper,

the uncertainty of an estimate is termed the standard

deviation and labeled sd[estimate]. Also used is the

coefficient of variation that is defined as cv[esti-

mate] = sd[estimate]/|estimate|. When this fitting

process is applied to the six independent elements of

the dynamic moment rate spectral modulus (Fig. 8)

for event 12 of Table 1, the results listed in Table 2

are obtained.

Table 2

Parameters of the dynamic moment rate spectral moduli for event 12

Parameter _M11_M21

_M22_M31

_M32_M33

~mij (GNm) 2,422.1 -2,447.4 2,106.1 874.6 1,841.2 -2,112.9

sd½ ~mij� (GNm) 979.5 993.3 866.3 360.1 733.4 874.2

cv½ ~mij� 0.40 0.41 0.41 0.40 0.40 0.41

fc (Hz) 10.2 6.7 11.5 9.2 8.1 11.7

sh 3.0 2.9 3.4 2.8 3.0 3.4

The static moment tensor elements ~mij and their uncertainties are estimated from the frequency interval between 1 Hz and the corner

frequency fc

Vol. 171, (2014) Source Mechanisms of Induced Earthquakes 1651

A notable feature of Table 2 is the fact that the

corner frequency fc is considerably larger on the

diagonal elements than on the off-diagonal elements

of the moment tensor. To determine whether the

isotropic part of the moment tensor is causing this

effect, the fitting process is applied to the deviatoric

(Eq. 10) and isotropic (Eq. 9) parts of the moment

rate tensor and the results are listed in Table 3. This

-2

-1

0

1

2

3

4

Log(Frequency) (Hz)

Log(

Mom

ent R

ate)

(G

Nm

)

11

-1 0 1 2 3 -1 0 1 2 3 -2

-1

0

1

2

3

4

Log(Frequency) (Hz)

Log

(Mom

ent R

ate)

(G

Nm

)

21

-2

-1

0

1

2

3

4

Log(Frequency) (Hz)

Log(

Mom

ent R

ate)

(G

Nm

)

22

-1 0 1 2 3 -1 0 1 2 3 -2

-1

0

1

2

3

4

Log(Frequency) (Hz)

Log

(Mom

ent R

ate)

(G

Nm

)

31

-2

-1

0

1

2

3

4

Log(Frequency) (Hz)

Log(

Mom

ent R

ate)

(G

Nm

)

32

-1 0 1 2 3 -1 0 1 2 3 -2

-1

0

1

2

3

4

Log(Frequency) (Hz)

Log

(Mom

ent R

ate)

(G

Nm

)

33

Figure 8Spectral moduli of the moment rate tensor j _Mijðf Þj for event 12. The index ij is shown in the upper right of each panel. The solid line is the

estimated modulus, the long dash line is an estimate of its uncertainty, and the short dash line is a model fitted to the modulus

1652 L. R. Johnson Pure Appl. Geophys.

table clearly shows that the corner frequency of the

deviatoric moment tensor is significantly less than

that of the isotropic part. Averaging the six elements

of the deviatoric moment tensor gives f cd ¼ 7:7� 0:9

Hz, whereas for the isotropic part. fci = 13.3 Hz. The

high-frequency power is also smaller for the devia-

toric part and has the average shd ¼ 2:9� 0:1

compared to shi = 3.5 for the isotropic part. The

coefficient of variation for the static moment is

somewhat less in Table 3 than in Table 2, indicating

that the spectral modulus of the deviatoric moment

tensor is a slightly better fit than the complete

moment tensor. These results show that for this event

the isotropic part of the moment tensor is enriched in

high frequencies with respect to the deviatoric

moment tensor.

The spectral moduli of the dynamic moment rate

tensor shown in Fig. 8 are only half of the solution.

The spectral phase is also available but not shown, as

it is difficult to display or parameterize the phase in a

meaningful way. However, the spectral phase plays a

critical role in determining the appearance of the

seismograms in the time domain.

When the spectral moduli of Fig. 8 are combined

with the associated spectral phase and transformed to

the time domain, the dynamic moment rate functions

of Fig. 9 are obtained. Although there is some acausal

signal before t = 0, a prominent pulse begins at about

this time on all elements of the moment rate tensor.

The shape of this first pulse, which lasts for about 0.1

s, is quite similar on all six elements of the moment

rate tensor, but this similarity degrades at larger

times. The diagonal elements show an increase in

similarity beginning at about 1 s. The duration of

significant signal is about 0.8 s on the off-diagonal

elements, but about 2.4 s on the diagonal elements.

These differences in signal between the diagonal and

off-diagonal elements of the moment rate function

add support to the differences in frequency content

found in Tables 2 and 3 and the idea that the devia-

toric and isotropic moment tensors are significantly

different.

The dynamic moment rate tensor elements in

Fig. 9 can be integrated to obtain the dynamic

moment tensor elements in Fig. 10. This integration

process is complicated by the uncertainty in the

moment rate tensor at low frequencies that was dis-

cussed earlier with respect to Fig. 8. This difficulty

can be mitigated somewhat by using Eq. 11 and

setting the long-time level of the moment function

Table 3

Parameters of the deviatoric and isotropic dynamic moment rate spectral moduli of event 12

Parameter _M011_M021

_M022_M031

_M032_M033

_Miso

~mij (GNm) 2,088.4 -2,447.4 1,574.0 874.6 1,841.2 2,739.8 1,374.3

sd½ ~mij� (GNm) 805.4 993.3 448.7 360.1 733.4 1,038.0 602.3

cv½ ~mij� 0.38 0.41 0.28 0.40 0.40 0.39 0.44

fc (Hz) 7.3 6.7 8.5 9.2 8.1 6.8 13.3

sh 2.7 2.9 2.9 2.8 3.0 2.9 3.5

The static moment tensor elements ~mij and their uncertainties are estimated from the frequency interval between 1 Hz and the corner

frequency fc

-1 0 1 2 3

Time (sec)

11 0.24E+05

21 0.20E+05

22 0.23E+05

31 0.79E+04

32 0.15E+05

33 0.16E+05

Figure 9The dynamic moment rate tensor _mijðtÞ that is obtained as inverse

Fourier transforms of the spectral moduli in Fig. 8 along with the

associated spectral phases. The numbers on the right are maximum

amplitudes in GNm/s

Vol. 171, (2014) Source Mechanisms of Induced Earthquakes 1653

equal to the low-frequency level of the moment rate

functions as determined by ~mij of Eq. 14. However,

there still exists a long-time drift in some of the

elements that is most likely related to the low-fre-

quency problem. To a first approximation, the

dynamic moment tensors of Fig. 10 are step functions

with finite rise times. Superposed on this simple time

function are oscillations that vary between the vari-

ous elements, Particularly conspicuous is the

similarity in the waveforms between 1 and 2.4 s on

the three diagonal elements that is not shared with the

off-diagonal elements.

The moment rate tensor of Fig. 8 can be substi-

tuted on the right hand side of Eq. 3b to obtain an

estimate of the calculated spectral modulus for the

three components of ground motion at station AL1

for event 12. These spectra are shown in Fig. 11

along with the observed spectra at this station cor-

responding to the time traces in Fig. 7. There is fairly

good agreement between the observed and calculated

spectral moduli in the frequency range between about

1 and 100 Hz. At frequencies\1 Hz or[100 Hz the

agreement is not so good, but these are the frequency

ranges where the strength of the signal is the weakest.

This is consistent with the earlier discussion related

to Fig. 8 where it is concluded that only frequencies

between 1 and 100 Hz can be considered reliable.

Starting with the dynamic moment tensor of

Fig. 10 and using Eq. 2, leads to the calculated

seismograms that are shown in Fig. 12. Note that the

difficulties with low frequencies that are encountered

in obtaining the dynamic moment tensors of Fig. 10

are not apparent in Fig. 12 because any low fre-

quency inaccuracies become insignificant when the

instrument response is included. A comparison of

Figs. 7 and 12 reveals the differences between an

interpretation of the data that uses a static moment

tensor and one that uses a dynamic moment tensor.

The moment tensor used in the calculated traces of

Fig. 7 is constructed using the ~mij and average values

of fc and sh of Table 2. Figure 12 is constructed using

the dynamic moment tensors of Fig. 10. The same

Green function and instrument response is used in

calculating Figs. 7 and 12.

The degree of fit between the observed and cal-

culated seismograms in Figs. 7 and 12 is quite

different. The calculated results for the static moment

tensor in Fig. 7 do a reasonably good job of fitting the

direct p wave on the radial and vertical components

and the direct s wave on the transverse component,

but essentially ignore all other parts of the observed

seismograms. The calculated results for the dynamic

moment tensor in Fig. 12 also do a reasonably good

job of fitting the direct p wave on the radial and

vertical components and the direct s wave on the

transverse component, but in addition they do a much

better job of fitting the general pattern of amplitude

versus time on the seismograms over the entire time

interval. However, a close inspection of the phase of

the motion at any specific time reveals a rather poor

fit for much of the seismograms.

The comparison between observed and calculated

results in Figs. 7, 11 and 12 suggests an interpretation

of the reliability of the dynamic moment tensor

results. First, note that the calculated seismograms in

Figs. 7 and 12 are based on moment tensors that have

essentially the same spectral moduli of Fig. 8, the

latter being the estimated values and the former being

a model fitted to the estimated values. Furthermore,

Fig. 11 shows that the spectral moduli of the

observed seismograms are explained quite well by the

calculated results in the frequency range of

1–100 Hz. Thus, the difference between the calcu-

lated seismograms in Figs. 7 and 12 and the fact that

-1 0 1 2 3

Time (sec)

11 0.24E+04

21 0.24E+04

22 0.21E+04

31 0.96E+03

32 0.18E+04

33 0.21E+04

Figure 10The dynamic moment tensor mij(t) that is obtained by integrating

the moment rate tensor in Fig. 9. The numbers on the right are

maximum amplitudes in GNm

1654 L. R. Johnson Pure Appl. Geophys.

neither does a good job of fitting all of the observed

data must be attributed to the role of the spectral

phase. It appears that the spectral moduli of the

moment tensor are fairly well determined, but the

associated spectral phases contain considerable

uncertainty. The sources of uncertainty in the

moment tensor inverse problem, such as the noise in

the data, the errors in the source location, and the

inaccuracies in the Green functions, can all be

regarded as noise that gets transformed primarily into

the spectral phase of the solution. This interpretation

is consistent with the description of random data that

have a deterministic power spectral density function

with all of the randomness contained in the associated

phase function (BENDAT and PIERSOL 1971). This line

of reasoning leads to the conclusion that in this study

the spectral moduli of the dynamic moment rate

tensors are reliably determined in the frequency range

between 1 and 100 Hz, but the same claim cannot be

made for the spectral phases.

These questions about the reliability of the spec-

tral phases of the dynamic moment rate tensor lead to

somewhat of a quandary when it comes to interpret-

ing the dynamic moment tensor in Fig. 10. Some of

the temporal features in the tensor elements may be

real properties of the source and some may just be

artifacts, but it is difficult to reliably determine which

is the proper interpretation of any particular feature.

The fit to the direct p wave on the R and Z compo-

nents and direct s wave on the T component in

Fig. 12 suggests that the initial 0.1 s of the dynamic

moment tensor may be a reasonably well resolved

feature. The long time level of the dynamic moment

rate tensor, which is based on the low frequency

R

-1

0

1

2

3

4

5

Log(Frequency) (Hz)

Log

(Am

plitu

de)

(m

/sec

)

T

-1

0

1

2

3

4

5

Log(Frequency) (Hz)

Log

(Am

plitu

de)

(m

/sec

)

Z

-1 0 1 2 3 -1 0 1 2 3

-1 0 1 2 3 -1

0

1

2

3

4

5

Log(Frequency) (Hz)

Log

(Am

plitu

de)

(m

/sec

)

Figure 11Spectral moduli of the observed data (solid line) and the calculated data (dashed line) for the three components of ground motion at station

AL1 for event 12

Vol. 171, (2014) Source Mechanisms of Induced Earthquakes 1655

spectral level of the moment rate tensor, is also well

resolved. However, in the intervening interval the

moment tensor elements show considerable oscilla-

tion and the reliability of these features is highly

suspect.

6. Static Moment Tensor Results

While the dynamic moment tensor gives the most

complete representation of the source process with

the fewest number of assumptions, there are situa-

tions where a less complete description may be

sufficient. A static moment tensor can serve this

purpose. As pointed out where the static moment

tensor is defined in Eq. 11, there are a number of

different methods of calculating the static moment

tensor. In the present study this is a useful property

because a comparison between different types of

static moment tensors, and a comparison with the

dynamic moment tensor provides a check on the

validity of all the results.

One method of obtaining a static moment tensor is

to replace the complete seismogram with the mea-

sured amplitudes of a few particular phases such as

direct p or s waves. This approach is described and

tested with synthetic data in STUMP and JOHNSON

(1977). KIRKPATRICK et al. (1996) used this method to

obtain static moment tensors for a large number

earthquakes at The Geysers by measuring the

amplitudes of p and s wave pulses, correcting for the

instrument response, and then using ray theory to

calculate Green functions. That method is also used

in the present study with one modification. Instead of

using ray theory for the Green functions, the full

wave theory Green functions used in obtaining the

dynamic moment tensors are calculated and passed

through the instrument to produce synthetic seismo-

grams on which the amplitudes of p and s wave

pulses are measured. When static moment tensors

obtained using the original method of KIRKPATRICK

et al. (1996) are compared with those obtained with

the modified method of this study, the results are

almost identical in most cases, but in cases where

they differ, the modified approach gives the most

consistent results. The wave theory basis of the

modified approach has some advantages over the

original approach in the manner that near-field terms

and the instrument correction are handled. When this

modified approach is applied to the data of event 12

of Table 1 the first two types of static moment tensors

listed in Table 4 are obtained. The first type uses only

the amplitudes of p waves measured on vertical

component seismograms, while the second type uses

both the amplitudes of p waves on vertical compo-

nent seismograms and the amplitudes of s waves on

transverse component seismograms. Other combina-

tions of phases and components could also be used.

Solutions for these types of static moment tensors are

straightforward least squares estimates so standard

errors of the estimates are available, and these

uncertainties are also listed in Table 4.

It is also possible to extract static moment tensors

from the dynamic moment tensor. O’CONNELL and

JOHNSON (1988) first calculated dynamic moment

tensors for three events at The Geysers and then

averaged these results over a time interval of 0.5 s to

obtain an estimate of the static moment tensor. The

same method is used in this study and the dynamic

moment tensors of Fig. 10 are averaged over the time

interval 0–0.1 s to arrive at the estimates that are

listed as type 3 in Table 4. The standard error of the

mean is also listed as an estimate of the uncertainty.

0 1 2 3 4

Time (sec)

R 0.63E+05

0.45E+05

T 0.56E+05

0.49E+05

Z 0.47E+05

0.21E+05

Figure 12Similar to Fig. 7 for the observed seismograms at station AL1 for

event 12. However, in this case the 2nd, 4th, and 6th traces are the

calculated seismograms that result when the dynamic moment

tensor of Fig. 10 is combined with the Green function and

instrument response

1656 L. R. Johnson Pure Appl. Geophys.

With this type of static moment tensor the results

depend upon the time interval that is selected for

averaging, which follows from the fact that the six

independent elements of the dynamic moment tensor

shown in Fig. 10 do not have identical time functions.

Another method of obtaining a static moment

tensor from the dynamic moment tensor is to use the

parameter ~mij of the spectral model given in Eq. 14.

This type of estimate and its uncertainty are already

listed in Table 2. This is basically an estimate of the

level of the spectral modulus at low frequencies, and

in this study that turns out to be the frequency interval

between 1 Hz and the corner frequency fc. The static

moment tensor of this type is listed as type 4 in

Table 4.

A common method of displaying the information

contained in a static moment tensor is to calculate

the eigenvalues (Eq. 7) and eigenvectors (Eq. 8) and

then plot the T, I, and P axes on a stereographic

projection of the focal sphere. This is done in

Fig. 13 for the four different types of static moment

tensors listed in Table 4. The basic information

contained in these plots is the points where the T, I,

and P axes pierce the lower half of the focal sphere.

The solid lines on the plots are the intersection with

the focal sphere of the planes orthogonal to the T

and P axes. In addition, the stippled areas on the

plots show the areas where P waves emerging from

the source have a polarity away from the source,

which corresponds to upward motion on a vertical

component seismogram.

The results listed in Table 4 and displayed in

Fig. 13 provide a sample of the variety that is pos-

sible in estimating static moment tensors. Type 1

(Fig. 13a) and type 2 (Fig. 13b) are derived from the

amplitudes of discrete phases on the seismograms,

whereas types 3 and 4 are derived from the dynamic

moment tensor that is obtained from analysis of

complete seismograms. Type 3 (Fig. 13c) is obtained

from a short time interval at the beginning of the

dynamic moment tensor while type 4 (Fig. 13d) is

obtained from a broad frequency interval of its

spectral modulus. Table 4 shows that all four types

have the same general pattern in terms of the signs of

the moment tensor elements, but the amplitudes of

the elements show considerable variation. The first

three types are based on the energy that is contained

in relatively short intervals of time and have ampli-

tudes that are much less than the fourth, which is

based on energy contained in the entire time interval

and a fairly broad frequency range. In spite of these

differences, the plots of principal axes shown in

Fig. 13 are very similar for the four different types of

static moment tensors.

Table 4

Different types of static moment tensors for event 12

Type ~m11 ~m21 ~m22 ~m31 ~m32 ~m33

p amplitudes

~mij (GNm) 645.9 -1,644.0 1,729.0 294.7 1,057.0 -1,166.0

sd½ ~mij� (GNm) 976.0 584.6 601.8 542.2 399.4 404.5

cv½ ~mij� 1.511 0.356 0.348 1.840 0.378 0.347

p and s amplitudes

~mij (GNm) 462.8 -1701.0 1752.0 744.8 447.5 1413.0

sd½ ~mij� (GNm) 12,170.0 2,965.0 11,740.0 2,110.0 2,747.0 10,580.0

cv½ ~mij� 26.296 1.743 6.701 2.836 6.138 7.488

Initial time

~mij (GNm) 657.6 -766.6 586.9 297.9 490.5 -602.2

sd½ ~mij� (GNm) 276.5 442.0 456.6 152.1 264.5 325.8

cv½ ~mij� 0.420 0.577 0.778 0.511 0.539 0.541

Low frequency

~mij (GNm) 2,422.1 -2,447.4 2,106.1 874.6 1,841.2 -2,112.9

sd½ ~mij� (GNm) 979.5 999.3 866.3 360.1 733.4 874.2

cv½ ~mij� 0.404 0.408 0.411 0.412 0.398 0.414

The third type uses the first 0.1 s of the time-domain dynamic moment tensor, and the fourth type uses frequencies between 1 Hz and the

corner frequency of the frequency-domain dynamic moment rate tensor

Vol. 171, (2014) Source Mechanisms of Induced Earthquakes 1657

On the basis of the experience acquired in esti-

mating the four different types of static moment

tensors of Table 4 and Fig. 13 for all of the events

listed in Table 1, some conclusions on the relative

reliability of the different types are possible. In

general the results for the 20 events are comparable to

those listed in Table 4 for event 12, but the plots of

the principal axes are not always as consistent as

those in Fig. 13. The method of the second type using

the amplitudes of p and s waves measured on the

seismogram is the least reliable of the four, primarily

because of difficulty in accurately measuring the

amplitudes of direct shear waves. The method of the

third type using the amplitudes of the dynamic

moment tensor averaged over a time interval has the

disadvantage that the results can depend upon the

selected time interval, which makes the results

somewhat subjective. The method of the first type

using the amplitudes of direct compressional waves

measured on the seismogram generally gives results

similar to the fourth type, but has the disadvantage of

depending on only a few data and is, thus, sensitive to

measurement errors. This first type also produces a

scalar moment that is considerably smaller than that

obtained with spectral methods. On the basis of this

experience, the method of the fourth type using the

low frequency level of the dynamic moment rate

tensor emerges as the most reliable and the preferred

method. It is based on a broad range of frequencies

and is derived from the analysis of complete three

N

S

W E+

+

- +

+

-

-

+

+

-

+

+

T

I

P

a N

S

W E+*

+*

-* +X

+X

-*

-X

+*

+*

-*

+X

+*

T I

P

b

N

S

W E

T

I

P

c N

S

W E

T

I

P

d

Figure 13Equal area stereographic projections of the principal axes of the four different types of static moment tensors listed in Table 4, with a

corresponding to type 1, b to type 2, c to type 3, and d to type 4. The positions of the T, I, and P axes are denoted and the two solid lines are the

planes perpendicular to the T and P axis. The areas where compressional waves leaving the source have a first motion in the away direction are

shown as stippled. For types a and b the points where rays penetrate the focal sphere are also shown

1658 L. R. Johnson Pure Appl. Geophys.

component seismograms. The relative uncertainties

for this approach are also the smallest and most

consistent.

The uncertainties in estimates of the static

moment tensor given in Tables 2 and 4 can be

propagated to the eigenvalues and eigenvectors using

first-order perturbation analysis (FRANKLIN 1968).

When this type of analysis is applied to the estimates

of ~mij and its uncertainties given in Table 2 for event

12, the results of Table 5 are obtained. Given the

results of Table 5, it is straightforward to further

propagate the uncertainties to the scalar moment

mo = 4,779.5 ± 482.6 GNm and the moment mag-

nitude Mw = 2.45 ± 0.04. Thus, the combination of

Tables 2, 3 and 5 gives a fairly complete description

of estimates with standard errors for the static

moment tensor, associated parameters, and basic

properties of the source time function for one par-

ticular event. Although there are some fluctuations,

the uncertainty in the source parameters of this event

can be roughly characterized by a coefficient of

variation of about 0.4.

7. General Results

The preceding two sections, Figs. 7 through 13,

and Table 2 through 5 outline the procedures used in

estimating moment tensors for one event, event 12 of

Table 1. Identical procedures are applied to the 19

other events in Table 1, and the results are similar to

those already described for event 12. With estimates

available for all of the events listed in Table 1, it is

possible to examine the similarities and differences of

these 20 events that cover a range in location, size,

and water injection rate. All of the static moment

tensors used in this section are estimated from the

low frequency level of the dynamic moment tensor

(type 4 of Table 4).

Some additional information about the events

selected for moment tensor inversion is listed in

Table 6. The events are selected so that the origin

times of the events spans the entire time interval from

212 days before until 267 days after the start of

injection into P32. The rate of injection at the time of

the event is listed in Table 6, as well the horizontal

distance of the hypocenter from the well P32. The

number of components used in the inversion K is

always at least 30. The stability of the dynamic

moment tensor inversion is measured by the condi-

tion number of the matrix Gðf Þ (Eq. 4), which is

approximated by the ratio of the maximum to

Table 5

Estimates of static moment tensor eigenvalues and eigenvectors for

event 12

K1;K2;K3 (GNm) 4,779.5 987.8 -3352.1

sd½K1;K2;K3� (GNm) 482.6 419.6 369.5

cv½K1;K2;K3� 0.101 0.425 0.110

e1T, e2

T, e3T 0.698 -0.709 -0.101

sd[e1T, e2

T, e3T] 0.188 0.172 0.188

cv[e1T, e2

T, e3T] 0.269 0.242 1.861

e1I , e2

I , e3I 0.646 0.563 0.516

sd[e1I , e2

I , e3I ] 0.216 0.226 0.223

cv[e1I , e2

I , e3I ] 0.334 0.401 0.432

e1P, e2

P, e3P -0.309 -0.426 0.850

sd[e1P, e2

P, e3P] 0.194 0.194 0.178

cv[e1P, e2

P, e3P] 0.628 0.455 0.209

Table 6

Information about events and dynamic moment tensor inversion

Event

no.

Time

(days)

Rate

(gal/min)

RP32 (m) K CNmin CNmax

1 -211.8 0 284.8 36 4.2 10.6

2 -164.4 0 717.0 39 3.2 7.9

3 -156.1 0 511.9 33 3.5 11.3

4 -85.8 0 586.0 36 3.7 11.9

5 -63.7 0 451.6 33 3.1 17.7

6 0.4 1,150 308.7 33 2.7 9.0

7 0.8 400 240.9 39 2.0 8.3

8 3.6 400 207.9 33 3.0 8.7

9 19.9 400 396.0 30 2.8 14.6

10 38.5 400 142.1 39 2.6 10.3

11 54.4 1,000 164.0 30 2.3 8.1

12 63.8 1,000 514.9 36 2.7 13.3

13 82.5 1,000 228.1 33 3.4 14.0

14 92.4 1,000 380.1 33 3.6 12.8

15 115.2 1,000 282.3 33 2.8 11.8

16 146.2 1,000 366.8 36 2.6 10.0

17 155.3 700 371.4 36 3.2 17.5

18 197.9 700 181.9 36 3.1 16.0

19 237.7 700 339.5 33 3.1 14.0

20 266.6 400 352.4 33 3.5 12.4

Time is measured in days after start of injection in P32, rate is the

injection rate of water into P32 at the time of the event, and RP32 is

the horizontal distance from P32 or its downward projection. The

number of components used in the moment tensor inversion is

given by K. The minimum and maximum condition numbers for

the inversion are CNmin and CNmax, respectively

Vol. 171, (2014) Source Mechanisms of Induced Earthquakes 1659

minimum eigenvalues (FRANKLIN 1968). This condi-

tion number is evaluated at each frequency, with the

minimum and maximum values for the entire fre-

quency range listed in Table 6. The maximum values

are always at frequencies \1 Hz or [100 Hz, with

the maximum values in the intervening interval are

never [10.

A property of interest about the moment tensors is

the relative size of the isotropic part of the moment

tensor, as this provides a measure of the volume

change in the source region. A discussion of the

isotropic part of the moment tensor must include the

possibility that it is an artifact, because it appears to

be the most poorly resolved part of the moment

tensor. Discussions of this matter can be found in

O’CONNELL and JOHNSON (1988), VASCO and JOHNSON

(1989), KUGE and LAY (1994), SILENY et al. (1996),

(2009), DUFUMIER and RIVERA (1997), JULIAN et al.

(1998) and COVELLONE and SAVAGE (2012). This

possibility of a greater uncertainty in the isotropic

part is supported by the results listed in Table 7,

where ~miso is considerably smaller than mo, but

sd½ ~miso� is only slightly smaller than sd[mo]. This

means that the relative error in ~miso is larger than that

in mo. Expressing this quantitively, the mean coeffi-

cient of variation cv[mo] as calculated from Table 7

is 0.11, while cv½ ~miso� is 0.25, which says that the

relative uncertainty of the isotropic part is over twice

that of the scalar moment. All of this uncertainty is

propagated throughout Table 7 so that meaningful

comparisons can be made. Note that O’CONNELL and

JOHNSON (1988) performed synthetic tests designed to

estimate the effect of source mislocation and incor-

rect velocity model on estimates of the isotropic part

of the moment tensor for earthquakes at The Geysers.

They found maximum fractional effects in the range

of 0.07–0.14 that are entirely consistent with the

estimates of sd½ ~miso=mo� listed in Table 7.

With these caveats in mind, consider the com-

parison of the isotropic part of the moment tensor to

the scalar moment in Table 7. The ratio of these two

quantities ~miso=mo ranges between -0.21 and ?0.47,

with the isotropic part negative for 4 and positive for

16 of the events. The mean value of this ratio for all

of the events is

~miso

mo

¼ 0:19� 0:10: ð15Þ

For the four events with a negative isotropic part, this

ratio has a mean of ~miso=mo ¼ �0:14 and an average

coefficient of variation of cv½ ~miso=mo� ¼ 0:87; while

for the 16 events with positive isotropic part these

Table 7

Comparison of scalar moment and isotropic moment

Event no. mo (GNm) sd[mo] (GNm) ~miso (GNm) sd½ ~miso� (GNm) ~miso=mo sd½ ~miso=mo� cv½ ~miso=mo�

1 467.1 48.9 52.8 40.0 0.11 0.09 0.76

2 199.9 27.0 -15.9 21.6 -0.08 0.11 1.34

3 388.4 33.9 60.5 29.4 0.16 0.08 0.49

4 36.8 3.9 10.9 3.9 0.29 0.11 0.37

5 218.2 27.8 37.0 23.0 0.17 0.11 0.75

6 24.8 2.9 4.3 2.6 0.17 0.10 0.61

7 541.4 60.4 253.3 50.9 0.47 0.11 0.23

8 148.1 13.0 28.0 13.4 0.19 0.09 0.49

9 2,805.6 217.9 -276.6 209.2 -0.10 0.07 0.69

10 412.8 57.0 -75.1 51.0 -0.18 0.13 0.81

11 312.8 29.7 71.6 28.6 0.23 0.09 0.41

12 4,779.5 482.6 805.1 426.4 0.17 0.09 0.54

13 3,985.7 426.8 710.2 411.4 0.18 0.11 0.59

14 13,154.3 1,507.5 2,065.8 1,516.1 0.16 0.12 0.74

15 133.1 18.5 -27.4 17.5 -0.21 0.13 0.66

16 1,539.4 155.5 312.2 143.1 0.20 0.09 0.47

17 1,776.9 212.2 478.5 199.4 0.27 0.12 0.43

18 175.5 14.4 42.4 13.5 0.24 0.08 0.33

19 16,712.0 2,058.2 1,815.5 1,720.9 0.11 0.10 0.95

20 1,602.1 184.2 300.3 157.7 0.19 0.10 0.54

1660 L. R. Johnson Pure Appl. Geophys.

numbers are ~miso=mo ¼ 0:21 and cv½ ~miso=mo� ¼ 0:54:

When individual events are considered, this ratio is

close to its uncertainty for some of the events. If one

accepts a coefficient of variation \1 as being signif-

icant, there are three events with negative isotropic

parts and 16 with positive isotropic parts. If one

accepts a coefficient of variation \0.5 as being sig-

nificant, there are no events with negative isotropic

parts and nine with positive isotropic parts, leaving

11 events to be classified as not having a significant

isotropic part.

The isotropic part of the moment tensor is related

to volume change in the source region by the formula

DV ¼ ~miso

j; ð16Þ

where j is the elastic bulk modulus. For the event

with the largest relative isotropic part, event 7, with

elastic parameters from Fig. 6 at the source depth, the

volume change is 6.45 m3. Considering all of the

earthquakes shown in Fig. 2 in the 300 days after the

start of injection, about 90 % of them have hypo-

centers within a volume of about 2.7 9 109 m3

centered near the bottom of P32. The cumulative

scalar moment of the events in this volume is

246 9 103 GNm and this gives a cumulative isotro-

pic part of 46.1 9 103 GNm if Eq. 15 is taken as a

valid average for all of these events. Then Eq. 16

gives 1.2 9 103 m3 as an estimate of the cumulative

volume change associated with earthquakes near P32,

and this is over six orders of magnitude smaller than

the volume containing the earthquakes. From Fig. 2,

the cumulative volume of water injected into P32 in

this same 300 days is 1.4 9 106 m3, and the volume

change associated with earthquakes is about three

orders of magnitude smaller than this number. Thus,

although the presence of an isotropic part in the

moment tensors of the induced earthquakes near P32

implies a cumulative increase in volume of over 103

m3, this amount is insignificant compared to the

volume containing the earthquakes or the volume of

water injected.

In Tables 2 and 3 it is shown that for event 12 the

corner frequencies and high frequency powers of the

moment rate spectral moduli are different for the

deviatoric and isotropic parts. That same pattern is

also present in the other 19 events in Table 1. In

Table 8 the average corner frequency of the six

devatoric elements of the moment tensor f0cd along

with its uncertainty sd½f 0cd� is listed and compared

with the corner frequency of the isotropic moment

tensor fci. The data are plotted in Fig. 14. Both corner

frequencies exhibit a tendency to decrease with

increasing scalar moment, although the scatter is

considerable. Linear regression yields

log10½f0cd� ¼ ð1:35� 0:08Þ � ð0:10� 0:03Þ log10½mo�

ð17Þ

for the average corner frequency of the deviatoric

moment tensor and

log10½fci� ¼ ð1:47� 0:08Þ � ð0:08� 0:03Þ log10½mo�ð18Þ

for the corner frequency of the isotropic part, with

both of these regression lines shown in Fig. 14. These

results indicate that, aside from the fact that the iso-

tropic corner frequency is somewhat larger, the two

corner frequencies have a similar dependence upon

scalar moment. The ratio of the two corner

Table 8

Spectral parameters of dynamic moment tensors

Event

no.

mo

(GNm)

f0cd

(Hz)

sd½f 0cd�(Hz)

fci

(Hz)

s0hd sd½s0hd� shi

1 467.1 18.1 7.7 23.3 3.5 0.7 4.0

2 199.9 15.9 1.5 25.2 3.3 0.3 4.3

3 388.4 15.9 4.4 24.1 3.5 0.5 4.6

4 36.8 15.8 1.5 18.4 2.3 0.2 2.9

5 218.2 9.3 2.4 15.5 2.2 0.2 3.1

6 24.8 11.1 1.5 21.2 2.2 0.2 3.2

7 541.4 9.8 1.0 12.3 2.9 0.1 3.2

8 148.1 11.4 1.7 16.7 2.7 0.1 3.2

9 2,805.1 9.8 1.6 16.1 2.9 0.1 3.5

10 412.8 15.5 2.2 18.7 3.3 0.2 3.9

11 312.8 15.9 2.8 23.4 2.9 0.2 3.7

12 4779.5 7.8 0.9 13.3 2.9 0.1 3.5

13 3,985.7 7.6 1.4 9.0 2.8 0.1 3.0

14 13,154.3 9.5 1.4 13.5 3.3 0.2 4.1

15 133.1 16.2 4.3 22.4 3.2 0.3 4.1

16 1,539.4 10.1 1.4 14.2 3.2 0.1 3.6

17 1,776.9 12.9 2.8 17.3 3.4 0.2 4.3

18 175.5 14.5 3.5 14.8 2.7 0.2 2.9

19 16,712.0 8.2 1.7 14.4 3.3 0.3 4.2

20 1,602.1 9.6 1.8 16.9 2.7 0.2 3.6

The corner frequencies f0cd and powers of the high frequency

asymptote s0hd are averages of the six elements of the deviatoric

dynamic moment rate tensor

Vol. 171, (2014) Source Mechanisms of Induced Earthquakes 1661

frequencies can also be considered. The dependence

on scalar moment is similar for the two so linear

regression yields

fci

f0cd

¼ 1:46� 0:23: ð19Þ

This result gives a quantitative measure of the dif-

ference in corner frequencies between the isotropic

and deviatoric parts of the moment tensor.

The same type of analysis can be performed for

the power of the high frequency asymptote. The

results are

log10½s0hd� ¼ ð0:36� 0:05Þ þ ð0:04� 0:02Þ log10½mo�ð20Þ

for the average high frequency slope of the deviatoric

moment tensor and

log10½shi� ¼ ð0:48� 0:05Þ þ ð0:03� 0:02Þ log10½mo�ð21Þ

for the high frequency slope of the isotropic part.

There is a slight tendency for the powers to increase

as the scalar moment increases. For the ratio of the

two powers

shi

s0hd

¼ 1:23� 0:09: ð22Þ

The power of the high frequency asymptote for the

isotropic part of the moment tensor is clearly larger

than that of the deviatoric moment tensor.

Figure 13 shows how stereographic projections

can be used to display the principal axes and isotropic

fraction of a static moment tensor. In Fig. 15 results

of this type are plotted next to the locations of the

epicenters for all of the events in Table 1. Some

spatial patterns are evident in this figure. A group of

eight events to the south of P32 (events 1, 3-7, 12,

18) all have a positive isotropic part and similar

orientations of the principal axes, the only varying

property being the value of the isotropic fraction. The

four events with negative isotropic parts (events 2, 9,

10, 15) all are located north of P32. Events 8 and 13

have near-by hypocenters and have similar mecha-

nisms, whereas event 18 is also nearby but has a quite

different mechanism. Event 7 has a mechanism

similar to event 18 and a similar epicenter, but has a

depth that is about 800 m above the other three

events. Events 11, 14, 17, and 20, which form a

cluster of events about 400 m east of the bottom of

P32, have very similar mechanisms, but event 19 is

about 200 m below this cluster and has a completely

different mechanism.

1 2 3 4 5 0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

Log[Scalar Moment], GNm

L

og(C

orne

r F

requ

ency

), H

z

Figure 14The corner frequency of the isotropic part of the dynamic moment

rate tensor fci (open circles and solid line) and the average corner

frequency of the deviatoric dynamic moment rate tensor f cd

(asterisks and dashed line) as a function of the scalar moment mo

on a log–log plot

-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

X (km)

Y (

km)

N

SW E

P32

1

2

3

4

5

6 7

8

9

10

11

12

13

14

15

16

17

18

19 20

Figure 15Map showing epicenters of events in Table 1 with attached equal

area stereographic projections for static moment tensors

1662 L. R. Johnson Pure Appl. Geophys.

Another way of comparing the principal axes that

are plotted in Fig. 15 is to plot them all together, and

this is done in Fig. 16 for the T and P axis. The T axes

are all close to horizontal with nine of the events

tightly clustered in the northwest-southeast direction.

These nine events (1, 3–7, 12, 17, 18) all have

positive isotropic parts (Table 7) and have an average

azimuth of N48�W. This common orientation of this

group of events is clearly displayed in Fig. 15 where

it is also seen that all in the group, except event 17,

are located south of P32. The P axes range from

horizontal to near vertical, which indicates both dip-

slip mechanisms (P axis near vertical) and strike-slip

mechanisms (P axis near horizontal).

8. Discussion and Conclusions

A general objective of this study is to investigate

source mechanisms of earthquakes at The Geysers by

solving for their first-degree dynamic moment ten-

sors. The results are first obtained in the frequency

domain, but are easily transformed to the time

domain. The major difficulty of this approach, as is

the case in many inverse problems, is to assess the

uncertainty of the dynamic moment tensors. There

are two primary sources of this uncertainty, the

uncertainty in the observational data and inaccuracies

in the Green functions that describe the wave prop-

agation effects. The limitations of the experimental

protocol, such as the instrument response or the

distribution of recording stations, are treated as part

of the uncertainty in the data.

The manner in which uncertainty in the observa-

tional data and Green functions gets transformed into

uncertainty in the dynamic moment tensors is best

considered in the frequency domain. There it is found

that the uncertainty is sufficiently large to prevent a

confident interpretation of the moment tensors for

frequencies \1 Hz and for frequencies [100 Hz. In

both cases there is a small signal-to-noise ratio

caused by a weak signal that is due to a fall-off in the

instrument response at low frequencies and a fall-off

in the source spectrum at high frequencies. The

conclusion is that the dynamic moment tensors of this

study are only reliable for frequencies between 1 and

100 Hz. Within this frequency interval, based on the

degree of fit between the observed and calculated

spectral moduli, it is concluded that the spectral

moduli of the dynamic moment tensors are estimated

with an uncertainty of about 10–50 %. However,

based on the lack of fit between the observed and

calculated time functions, it is concluded that most of

the uncertainty of the dynamic moment tensors

resides in the spectral phases. While some of the

features of the dynamic moment rate functions in the

time domain appear to be real, the reliability of other

features cannot be established.

Even when restricted to the use of the spectral

moduli in a limited frequency range, the estimation of

dynamic moment tensors is still an effective method

of investigating seismic source mechanisms. This

N

S

W E

T

T

TTTT

T

T

T

T

TT

T

T

T

T

T

T

T

T

a N

S

W E

P

P

P

P

PP

P

P

P

P

P

P

P

P

P

P

P

P

P

P

b

Figure 16Equal area stereographic projections of the a T axes and b P axes on the lower half of the focal sphere for the 20 events in Table 1

Vol. 171, (2014) Source Mechanisms of Induced Earthquakes 1663

approach has the advantage that it is based on

essentially all of the information contained on the

seismograms and does not require any subjective

identifications and measurements of specific seismic

phases. The spectral moduli of the dynamic moment

tensors are easily parameterized in a manner that

provides quantitative information about the source

mechanism. Of the different types of static moment

tensors investigated in this study, it is concluded that

the most reliable is based on the low frequency level

of spectral moduli from dynamic moment rate func-

tions. The corner frequency of the source time

function and the power of its high-frequency

asymptote can also be determined with this approach.

Many of the results obtained in this study are

consistent with those of other studies. KIRKPATRICK

et al. (1996), ROSS et al. (1996, 1999) and GUILHEM

et al. (2014) all found about half of the events at The

Geysers to have significant isotropic parts, both

positive and negative, with the fraction as large as

0.3. The results listed in Table 7 are generally con-

sistent with these other studies, with the isotropic

fraction as large as 0.47 in one case. However, when

uncertainties are also considered, conclusions about

the fraction of events with significant isotropic parts

and the existence of events with negative isotropic

parts depend upon the level of significance that is

desired. Both OPPENHEIMER (1986) and KIRKPATRICK

et al. (1996) found the T axis of focal mechanisms to

be predominantly horizontal, and this agrees with

Fig. 16a.

The parameterized spectral modulus of the

dynamic moment tensor provides useful information

about the corner frequency and the power of the high

frequency asymptote of the source time functions.

The results of this study suggest that there can be

significant differences between these parameters for

the isotropic part and the deviatoric moment tensor.

The finding that the average corner frequency of the

isotropic part is more than 40 % larger than that of the

deviatoric moment tensor is a significant difference.

An explanation could simply be that the isotropic part

is primarily determined by p waves, while the devi-

atoric moment tensor is primarily determined by s

waves. SATO and HIRASAWA (1973) developed a

kinematic failure model of a circular shear crack and

found that the ratio of the corner frequency of

radiated p waves to radiated s waves varied between

1.26 and 1.39 as the rupture velocity varied between

0.5 and 0.9 times the shear velocity at the source.

The estimated corner frequencies are also affected

by the models of attenuation used in the moment

tensor inversion, and this could play a role in deter-

mining the ratio of the isotropic to deviatoric corner

frequencies. A series of tests were performed for six

of the events listed in Table 1 (1, 7, 9, 12, 14, and 19)

in order to explore this possibility. First, the Qp val-

ues of the velocity model were increased by a factor

of two to Qp = 200 above sea level and Qp = 400

below sea level and the entire inversion process

repeated. The primary effect of this change is to

increase the power of the high frequency asymptote

for the isotropic part so that there is a slight increase

in the corner frequencies of both the isotropic and

deviatoric parts of the dynamic moment tensor with

little change in their ratio. Next, the Qs values of the

velocity model were decreased by a factor of two to

Qs = 50 above sea level and Qs = 100 below sea

level and the entire inversion process repeated once

again. The change in this case causes an increase in

the corner frequency f0cd of the deviatoric part, but

there is also an increase in the corner frequency of the

isotropic part fci. The net result is a ratio of the iso-

tropic to deviatoric corner frequencies that is reduced

to near 1.0 for four of the events (1, 7, 9, and 14) but

remains at approximately 1.4 for the other two events

(12 and 19). Thus, the cause of the observed differ-

ence between the isotropic and deviatoric corner

frequencies remains uncertain, with possible expla-

nations including a natural effect of the source failure

process, an incorrect treatment of attenuation, or it

could be that the isotropic and deviatoric parts are

produced by two different source processes.

The focal mechanisms shown in Figs. 15 and 16

contain two somewhat contradictory features. Groups

of events contained in a certain spatial area have very

similar focal mechanisms, both with respect to the

orientation of the principal axes, and the sign of the

isotropic part. At the same time, close to some of the

members of such a group there are other events that

have completely different focal mechanisms. Further

investigation is needed to determine if these spatial

patterns are related to features of the geology or

hydrology.

1664 L. R. Johnson Pure Appl. Geophys.

A matter not yet discussed is the correlation

between the water injection rate and the properties of

the earthquakes selected for moment tensor inversion

(Fig. 2). The reason for this is that no specific cor-

relation has been detected. There appear to be no

recognizable differences between the moment tensors

of earthquakes that occurred before injection began

and those that occurred after injection began, nor do

the moment tensors appear to change as the rate of

injection changes. These facts suggest that earth-

quakes occur whenever water is added at the

appropriate depth, with both the number and largest

size of the earthquakes being related to the amount of

added water. The small number of earthquakes that

occurred before the beginning of injection into P32

can be explained as due to natural recharge from rain

and streams and possibly from wells more distant

than P32. This type of reasoning is similar to that of

MAJER and PETERSON (2007) where the matter is dis-

cussed in more detail.

The careful examination of a large number of

seismograms recorded at The Geysers supports the

idea that scattering of elastic waves plays an impor-

tant role in determining the appearance of these

seismograms. For instance, the observed seismo-

grams in Fig. 7 show many of the features considered

by SATO et al. (2012) as indicative of high frequency

scattering by correlated random heterogeneity,

including the codas of P and S waves that last longer

than the expected source duration, an S coda that is

more prominent than that of P, and a tendency for

equipartition of motion on the different components.

The fact that the seismograms at each station have

certain distinctive characteristics, such as frequency

content, shape of phase arrivals, and duration of

phases, that are independent of the source suggests

that much of this scattering is taking place near the

seismic station. This idea is consistent with the

problems encountered in estimating dynamic moment

tensors because scattering will have a minor effect

upon the modulus of the Fourier transform of the

seismogram but will have a major and largely random

effect upon the phase. It helps explain why ampli-

tudes of static moment tensors based on individual

phases are significantly smaller than those based on

the spectral modulus of the dynamic moment rate

tensor (Table 4). PATTON and AKI (1979) discuss this

matter of how various types of noise can cause

fluctuations in the spectral amplitudes and incoher-

ency in the spectral phases of the moment tensor.

Reasons for scattering of high frequency elastic

waves at The Geysers can be found in the topography

and heterogeneity in material properties. Numerical

simulations have shown that topography such as that

found at The Geysers can cause variations in the

amplitudes of seismic waves of more than 50 % at

frequencies above 2 Hz (BOORE 1972, 1973; BOUCHON

et al. 1996). The effects of heterogeneity in velocity

has also been investigated with numerical calcula-

tions (FRANKEL and CLAYTON 1986; O’CONNELL 1999;

SATO et al. 2012) and effects similar to those shown

in Fig. 7 can be produced by velocity variations of

only about 5 %. The effects of heterogeneity are

complicated and involve the types of correlation

functions used to describe the velocity variations, the

variance and correlation length of velocity variations,

the relative roles of forward and backward scattering,

and whether the scattering is single or multiple. The

properties of the attenuation that accompanies the

scattering are particularly sensitive to these details of

the scattering process. These results suggest that

understanding the cause of uncertainty in the moment

tensor estimates may require a thorough character-

ization of scattering in the shallow crust at The

Geysers, a fairly difficult task that is beyond the

scope of the present study.

If scattering plays the significant role that is

suggested, then some of the inaccuracy in the Green

functions may be related to the manner in which

attenuation is included, as the phase responses of

intrinsic attenuation and scattering attenuation are

generally different. Uncertainty in the type and values

of attenuation included in the Green functions will

also have an effect upon the corner frequencies and

high frequency powers that are listed in Table 8.

Three-dimensional variations in the attenuation may

also be important. ROMERO et al. (1997) found that

frequency dependent attenuation corrections had to

be made for each recording site at The Geysers.

The testing and validation of the results of this

study will require comparisons with data from other

disciplines such as geology and hydrology. Given

that injection of water causes an increase in the

number of earthquakes (Fig. 2), the presence and

Vol. 171, (2014) Source Mechanisms of Induced Earthquakes 1665

pressure of liquids and gases plus changes in tem-

perature may all play a role in the earthquake process.

Further effort is also required to help establish that

moment tensor results obtained in this study of 20

events are truly representative of the over 2,000 other

events that occurred in the same study area in the

same time interval. Finally, a better understanding of

the results could benefit from the development of a

physical model of the earthquake source process that

is appropriate for these types of induced earthquakes,

and that is the subject of a companion study (JOHNSON

2014).

Acknowledgments

This study benefited greatly from the accumulated

knowledge and broad expertise concerning The

Geysers geothermal reservoir that exists at LBNL.

Those who were particularly helpful include Ernie

Majer, John Peterson, Don Vasco, Larry Hutchings,

Gisela Viegas, Katie Boyle, Hunter Philson, and

Aurelie Guilhem. The comments of two anonymous

reviewers helped to improve the manuscript. The

research is supported by the Department of Energy

Office of Basic Energy Sciences and is conducted at

LBNL, which is operated by the University of

California for the US Department of Energy under

Contract No. DE-AC02-05CH11232.

Open Access This article is distributed under the terms of the

Creative Commons Attribution License which permits any use,

distribution, and reproduction in any medium, provided the original

author(s) and the source are credited.

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(Received May 29, 2013, revised January 30, 2014, accepted February 8, 2014, Published online April 19, 2014)

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